Warped Product Pointwise Hemi-Slant Submanifolds of Nearly Kaehler Manifolds

In this paper, we introduce the notion of pointwise hemi-slant submanifolds of nearly Kaehler manifolds. Further, we study their warped products and prove the necessary and sufficient condition that a pointwise hemi-slant submanifold to be a warped product manifold. Also, we prove that every pointwise hemi-slant warped product submanifold $M=M_\perp\times_fM_\theta$ which is mixed totally geodesic in an arbitrary nearly Kaehler manifold $\tilde M$ satisfies $\|h\|^2\geq\frac{2p}{9}\cos^2\theta\|\nabla(\ln f)\|^2,$ where $\|h\|$ is the length of the second fundamental form of $M$ and $2p=\dim M_\theta$; while $\nabla(\ln f)$ is the gradient of $\ln f$ along $M_\perp$. The equality case of this inequality is also given.